Distributions for Waiting Times

Under the following assumptions, Yt number of occurrences in the time interval (0, t) has the Poisson distribution with mean p, = Xt. 

Occurrences of an event happen randomly through time at a constant rate A. 

What happens in one time interval is independent of what happens in another non-overlapping time interval, independent increments) 

The probability of exactly one event occurring in a very short time interval (f, t + h)is approximately proportional to the length of the interval h. 

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Sometimes the random variable we are interested in is the time we have to wait until some event occurs. In this section we look at four distributions for waiting time, two discrete distributions associated with Bernoulli processes, and two continuous distributions associated with the Poisson process. In each situation the first distribution is a special case of the second. 

Many commonly used distributions are members of the one-dimensional exponential family. These include the binomial n, n) and Poisson fi) distributions for count data, the geometric ir) and negative binomial r, tt) distributions for waiting time in a Bernoulli process, the exponential ) and gamma n, A) distributions for waiting times in a Poisson process, and the normal p, o ) where [Pg.89]

Here we investigate the distribution of waiting times for the event that at least one channel is open for a certain time. We also calculate the probability for a single... 

For every individual simulation process, i.e. for each size of channel cluster, the waiting time for the event at least one channel is open for f/ was detected this was repeated 400 times. Then the distribution of waiting times, w (t) was estimated from binned data. An example of a histogram obtained is seen in Fig. 11.3 for a cluster that contains three channels, with ma = 1.9 ms and ms = 5 ms. 

One of the aspects associated with vacancy-mediated diffusion that differentiates it from the hopping mechanism on this surface is the long waiting time between consecutive jumps. An example of the distribution of waiting times has been plotted for both In and Pd in Fig. 5. 

Figure 5 Distributions of waiting times between subsequent jumps in Cu(0 01), measured for (a) In at 320 K and (b) Pd at 335 K. Both fits are pure exponentials. The time constant t is shown in the graphs for both distributions.

In general, three steps are required to create trajectories by means of RW simulations,76,84 (i) determination of an initial orientation/position so that the equilibrium distribution is obeyed, (ii) random selection of a waiting time tw between two subsequent jumps from a suitable distribution and (iii) calculation of the new orientation/position after the jump. After step (i), the steps (ii) and (iii) are performed alternately until a trajectory of sufficient length in time is obtained. While the time scale of the motion is determined by the distribution of waiting times g(tw) in step (ii), the geometry of the motion comes into play in step (iii). For example, the new orientation 0i+ after a -degree-rotational jump of a C-2H bond can be obtained from the old orientation 0, according to... 

In glassy polymeric media, electronic transport can be phenomenologically thought of as a series of discrete steps characterized by a distribution of waiting times i /(t). When the distribution extends into the time scale of observation, the mobility itself will always appear to be time dependent. However, when the distribution does not extend into the time scale of observation, mobility can be characterized by an averaged value for most of the transit event, even though it exhibits thermalization at early times, which may be resolvable under certain experimental conditions (26, 27). Several more microscopically detailed pictures can correspond to this phenomenological description of electronic transport. 

L(t) will denote the load, or number of jobs in the system, waiting or being served, at time t 0. For each model considered in this section, the distribution of L(t) approaches a limit as t — and the limit is referred to as the steady-state distribution of the number in the system. (As we said in Subsection 3.3, we do not distinguish carefully between steady-state and Kmiting distributions.) We denote by L a random variable whose distribution is this steady-state distribution. Let W denote the time-in-system or flow time of the nth arrival to the system. Then, for the models considered here, the distribution of W also has a limit as n —> , which we call the steady-state flow-time distribution. We denote by W a random variable with this distribution. In the same way we can define a random variable Wg whose distribution is the steady-state distribution of waiting time. Flow time and waiting time differ only in that the latter does not include service time. Most of the results given here concern the expected values of the random variables L, W, and Wg for various queues. These... 

The physical origin of a power law distribution function for waiting times might arise from an exponential distribution of activation energies. Suppose, the distribution function of activation energies was of the form... 

Let us come back now to the question of increasing heterogeneity in the local mobilities upon decreasing the temperature. We have already identified a tendency for immediate back jumps after one torsional transition as the reason for the different temperature dependencies of the mean waiting time between torsional transitions (twait) and the torsional autocorrelation time ttacf)- In a homogeneous system, where every chemically identical torsion shows identical dynamics on the time scales of observation the probability distribution of waiting times should be... 

Figure 9. Distribution of waiting times for a total of 10 torsional transitions per dihedral degree of freedom to occur plotted versus 10 times t/(twait)- The thick line is the Poisson distribution. Upon lowering the temperature the deviation from the Poisson distribution increases.

In reality, the queue size n and waiting time (w) do not behave as a zero-infinity step function at p = 1. Also at lower utilization factors (p < 1) queues are formed. This queuing is caused by the fact that when analysis times and arrival times are distributed around a mean value, incidently a new sample may arrive before the previous analysis is finished. Moreover, the queue length behaves as a time series which fluctuates about a mean value with a certain standard deviation. For instance, the average lengths of the queues formed in a particular laboratory for spectroscopic analysis by IR, H NMR, MS and C NMR are respectively 12, 39, 14 and 17 samples and the sample queues are Gaussian distributed (see Fig. 42.3). This is caused by the fluctuations in both the arrivals of the samples and the analysis times. 

Fig. 42.4. The ratio between the average waiting time (iv) and the average analysis time (AT) as a function of the utilization factor (p) for a system with exponentially distributed interarrival times and analysis times (M/M/1 system).

Fig. 42.7. Histograms and cumulative distributions of the delays (waiting time + analysis time) in a department for structural analysis. (I) Observed values. ( ) Cumulative distribution. ( ) Fit with a theoretical model (not discussed in this chapter).

The excited state of a molecule can last for some time or there can be an immediate return to the ground state. One useful way to think of this phenomenon is as a time-dependent statistical one. Most people are familiar with the Gaussian distribution used in describing errors in measurement. There is no time dependence implied in that distribution. A time-dependent statistical argument is more related to If I wait long enough it will happen view of a process. Fluorescence decay is not the only chemically important, time-dependent process, of course. Other examples are chemical reactions and radioactive decay. 

The outflow of a CSTR is a Poisson process, i.e., fluid elements are randomly selected regardless of theirposition in the reactor. The waiting time before selection for a Poisson process has an exponential probability distribution. 

Figure 24 Probability distributions for the waiting time for 10 dihedral transitions. Time is given in units of the average waiting time 10x. The distributions are peaked around 10 = 1 and are much broader than the Poisson distribution but approach it for high T. For low T, a high probability for short waiting times exists and a long time tail of the distribution develops.

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

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